Mathematical Physics Vol 1

7.3 Linear and quasilinear first order PDE

347

7.3.6 Pfaffian equation

Pfaffian equation is the equation of the form

P d x + Q d y + R d z = 0 ,

(7.36)

where z = z ( x , y ) is the unknown function, and P , Q and R are given, continuously differentiable functions P = P ( x , y , z ) , Q = Q ( x , y , z ) , R = R ( x , y , z ) , in 3–dimensional space R 3 . The equation (7.36) can be relatively easily integrated in two cases: 1 ◦ when the left hand side represents a total differential of some function, which we shall denote by u , and 2 ◦ when there is such a function (integration factor) by which the Pfaffian equation needs to be multiplied to obtain a total differential. In the first case, observe some function u ( x , y , z ) , whose total differential is

∂ u ∂ x

∂ u ∂ y

∂ u ∂ z

d u =

d x +

d y +

d z .

By comparison with (7.36), in order to be a total differential, this expression must satisfy the following conditions

∂ u ∂ x

∂ u ∂ y

∂ u ∂ z

= P ,

= Q ,

= R .

(7.37)

If u is a twice continuously differentiable function ( u ∈ C 2 ( D ) ), then ∂ 2 u ∂ x ∂ y = ∂ 2 u ∂ y ∂ x , ∂ 2 u ∂ y ∂ z = ∂ 2 u ∂ z ∂ y , ∂ 2 u ∂ z ∂ x = ∂ 2 u ∂ x ∂ z .

(7.38)

From (7.37) and (7.38) we obtain ∂ ∂ x ( Q )= ∂ ∂ y ( P ) ,

∂ ∂ y

∂ ∂ z

∂ ∂ z

∂ ∂ x

( R )=

( Q ) ,

( P )=

( R ) ,

(7.39)

the integrability conditions for the initial equation (7.36). Thus, in this case, the equation (7.36) can be written as d u = P d x + Q d y + R d z = 0 (7.40) from where, by integrating, we obtain the implicit solution of the initial equation u ( x , y , z )= Z x x 0 P d x + Z y y 0 Q d y + Z z z 0 R d z = c , (7.41) where c is an arbitrary constant. In the second case, assume that there exists a function v ( x , y , z ) , such that d u , defined by the relation d u = vP d x + vQ d y + vR d z = 0 is a total differential. Let us now look for the conditions that the functions P , Q and R must satisfy in order for the function v to exist, as well as for the relation from which we could determine the function v . Similarly to the previous case, we have the conditions

∂ u ∂ x

∂ u ∂ y

∂ u ∂ z

= vP ,

= vQ ,

= vR ,

(7.42)